Causal monotonicity, omniscient foliations and the shape of space (Q2704291)

The authors of this article are interested in clarifying what one might mean by the phrase ``shape of space'' in the setting of classical general relativity. Consequently, they give a definition of shape of space and state conditions that guarantee that a spacetime has a well-defined shape of space. By shape of space, the authors roughly mean the diffeomorphism class of sufficiently well-behaved edgeless spacelike hypersurfaces in the spacetime, in the situation when there is only one such class. In order to show that a large class of spacetimes exist which have only one such diffeomorphism class, the authors study in some detail spacetimes foliated by timelike curves. The idea is that under suitable conditions, which they state precisely, the quotient space formed by identifying points on the same timelike curve yields a shape of space. In the case of immersed spacelike hypersurfaces in strongly causal spacetimes, they state conditions based on the action of the immersion on the fundamental group that guarantee an actual shape of space.NEWLINENEWLINENEWLINEThis well-written article includes many specific examples that help clarify the technical conditions. The work is based on a series of papers written by one of the current authors (S. G. Harris) and several other collaborators over the past thirteen years. In particular, it is an improvement of several results announced in the article by the two current authors [\textit{S. G. Harris} and \textit{R. J. Low}, Geometry and Topology of Submanifolds Vol. VII, Differential Geometry in Honour of Professor Katsumi Nomizu, F. Dillen (ed.) et al, 136-138 (1995; Zbl 0841.53001)] and an article by \textit{S. G. Harris} [Proc. Sympos. Pure Math. 54, Part 2, 287-296 (1993; Zbl 0796.53066)].